Excerpt from the Proceedings of the COMSOL Conference 21 Boston Uncertainty Analysis, Verification and Validation of a Stress Concentration in a Cantilever Beam S. Kargar *, D.M. Bardot. University of Alabama in Huntsville *N274 Technology Hall, Huntsville, AL 35899, sk6@uah.edu Abstract: Stress concentration is an important effect in structural mechanics produced by a concentrated load or discontinuity in the geometry of a structure; stress concentration factor is the ratio of raised stress and nominal stress. In this paper we compare the results from a bending cantilever beam experiment with the theoretical values and COMSOL finite element simulation. In the experiment a simple cantilever beam with a hole is loaded at the end. This geometric discontinuity causes a stress concentration around the hole and this stress concentration is reported with estimates of the experimental uncertainty. To determine the optimum mesh size for the simulation, a grid convergence study was conducted by monitoring the predicted maximum deflection at the end of the beam and comparing it to the theoretical value. Monte Carlo technique was used to quantify the uncertainties associated with the simulation inputs such as the measured beam dimensions and modulus of elasticity. This paper presents an end-to-end example of uncertainty quantification, model verification and validation. Keywords: Stress concentration, strain cantilever beam, random and systematic uncertainty. 1.1 Experiment Setup The material used for the experiment includes a high strength aluminum beam with the hole and the strain gages attached. Measurements are recorded by four strain gages mounted on the beam, three near the hole and one at a location where the nominal stress is equivalent to that of a similar beam without a hole. The strain gage types for the three small sensors next to the hole are called G1388 by the manufacturer (VISHAY measurements group) which have grids no longer than.31.31 inch, and the forth one is a 125AD strain gage which can be bigger with a gage length of about.25 inch. For displaying the strain while loading, we used a digital strain indicator P-35 made by the same company. The model P-35 strain indicator is a portable, battery-powered precision instrument for use with resistive strain gages and transducers. The P-35 will accept full, half, or quarter-bridge inputs 2. Strain gages are normally connected via a front-panel binding posts. The beam is loaded simply with a hanger and four weights each one pound with a total of four pounds plus the weight of the hanger. Figure 1 shows the set up for the experiment and the strain gages on the beam. 1. Introduction In this paper we looked at the stress concentration in a loaded cantilever beam which has a through hole. The Structural Mechanics Module in COMSOL version 4.a was used to create a model of this beam. Simulations of this loaded beam are compared with theoretical values and experimental results. Also presented are the uncertainties both in simulation and experimentation. The Monte Carlo method was used as the method to estimate the total uncertainty which includes random and systematic contributions. 1 Figure 1. The experiment set up, strain indicator, beam in loading (upper). The beam with the hole and strain gages attached (lower).
1.2 Simulation in COMSOL The geometry of the beam was created in COMSOL using the CAD device. The beam is made in 3-D space according to the nominal dimensions of the actual beam for the experiment. The fixed end is constrained and the end load is applied as a load per length. Two sets of cut-lines are defined to indicate the position of readings for strain, one through the hole and one at the nominal strain. Figure 1 shows the cutlines in 3D. In order to find the optimized tetrahedral mesh size for the simulations we conducted grid convergence study based on the stress concentration factor. For this purpose, nine different mesh sizes, from extremely coarse to extremely fine were used and the stress concentration factor is found for each case. Figure 3Figure 1 shows the stress concentration factor versus the number of elements produced in each case. 3 Table 1, the finer mesh size seems to be sufficient in that K t has converged to the second decimal place and the simulation time is not too onerous given the need to run thousands of simulation for the Monte Carlo Methods. Table 1. Different mesh sizes, stress concentration, number of elements, element quality and solution run time. Mesh Size Kt No of Elements Average Element Quality Solution Time (s) Extremely Coarse 1.1 84.2674 1.765 Extra Coarse 1.292 12.3396 1.656 Coarser 1.197 191.582 1.563 Coarse 1.232 259.5874 1.593 Normal 1.718 644.77 1.734 Fine 1.842 134.7377 1.984 Finer 1.773 334.7746 2.5 Extra Fine 1.776 682.795 2.86 Extremely Fine 1.781 168.8213 4.422 1.3 Simulation Verification Figure 2. 3D picture of the cut-lines, one through the hole and the other at the location of nominal strain (the location fourth strain gauge in the experiment). K t 2.3 1.8 1.3.8 Stress Concentration vs No of Elements 5 1 15 No. of Elements Figure 3. Stress concentration factor versus No. of elements; the finer the mesh size, the higher the No. of elements. In order to select the mesh size, we looked at the number of elements, average element quality, and solution time. In Table 1, K t is shown for each mesh case along with the time it took for the simulation to run. According to Figure 4 and For verification of our simulation, we compared the maximum deflection of the beam with the theoretical value for the first 5 cases of parametric study. The average value for simulation is equal to.142 ±.6 inch (± one standard deviation) and the average from the theory is equal to.141 ±.6 inch which is found from Equation 1: A beam was also modeled once without the hole and the end deflection is found from this simulation to be.138 ±.6 inch. Both simulations bracket the theoretical value of deflection. 2. Uncertainty in Experiment There are always errors and uncertainties in the experiments, most common ones stem from the equipment, environment and human repeatability. In order to find the uncertainty in experiment we need to investigate the equipment that we are using. The instruments for this experiment include the strain gages attached to
the beam and the strain indicator P-35. The accuracy for the gage factor as reported by the manufacture is ±1% and the resolution of the reading from the strain indicator is 1µε. Additionally the accuracy is ±.5% of the reading plus ±3µε for the gage factor 1. in MULT X1 position. (The MULT push button extends the measurement range of the instrument to ±199999µε when in the X1 position; the display reading must be multiplied by ten to obtain the correct numerical value. For this test the measurement range of ±19999µε is used, so we choose MULT X1). 2 Knowing this information about the equipment gives us the systematic uncertainty; In other words, for systematic uncertainty of the strain indicator, we generate random numbers with normal distribution with mean of zero and standard deviation of.5 (±.5% of the reading), and mean of zero and standard deviation of 3 (±3µε), and a uniform distribution of -.5 to.5 (1µε resolution); since all the strain measurements are made with the same device, we can simply consider them correlated which means the random numbers generated for the Monte Carlo samples for all the strain gages are exactly the same. For finding the random errors the experiment was repeated 1 times. The experiment requires the loading the beam with weights connecting each gage sequentially one at-a-time to the quarter Wheatstone bridge strain indicator and reading the initial and final strain; the net value is be calculated by subtracting the initial value from the final value. By having the strain readings for all four strain gages we can find the average and standard deviation for the strain measurement in µε. 1.654 1.652 1.65 1.648 1.646 1.644 Convergence of Kt for Experiment 1 2 3 Number of Samples Average Standard Deviation.17.16.15.14.13 Figure 4. Convergence of average and standard deviation of Kt for COMSOL simulation. For Monte Carlo simulation of the random error, random numbers are generated with a mean of zero and standard deviation equal to the experimental standard deviation. We run the Monte Carlo for 3 samples and compute K t, the concentration factor. Figure 4 shows the convergence of K t occurs at approximately 15 samples. Finally the uncertainty of K t is found by two methods; in the first method, the random and systematic uncertainty for K t is calculated as standard deviation and then they are combined together with Equation 2 as bellow, In the second method, the value of K t is found for all 3 samples including the random and systematic uncertainty and then the total uncertainty for K t is calculated by finding the standard deviation. 1 The average value of K t for both methods is equal to 1.646 and the uncertainty for method (1) and (2) which shows close results is: (1)=.141 (method 1) (2)=.157 (method 2) In Figure 5 the histogram of K t for the experiment is plotted. Number of Readings 4 2 Histogram of K t for Uncertainty in Experiment 1.584 1.595 1.67 1.619 1.631 1.642 1.654 1.666 1.678 Figure 5. Histogram of K t for the experiment 1.689 1.71 2.1 Stress Distribution around Circular Holes More Bin for K t Experiment Gaussian Distribution Discontinuity in geometry results in sudden changes in stress and strain. As we get far enough from the hole the stress distribution will be closer to the nominal value. In order to find the stress distribution we look at the differential form of equilibrium equation,
Where, X and Y, represent the components of body force. If we solve equations 3 and 4 for two dimensional problems by using the polar coordinate and assuming the body force equal to zero we will have, The forth strain gage,, is place where the stress is equivalent to the nominal value of stress if there was not any stress concentration. Since the strain gage factor is different for the three strain gages next to the hole and the forth one, we use by, Where, φ is the stress function as a function of r and θ. After applying the boundary condition for a circular hole in a plate subjected to uniform tension of S, the stress at the line across the hole and parallel to the width of the plate is (distributions can be viewed in Figure 6): 4 Since we are looking at the normal stress distribution on the surface of a cantilever beam, where the shear stress is zero, we can use the derived equations from the uniform loaded plate. Finally, the equation we will use for stress distribution is, Where, R is the radius of the circular hole and Z i is the distance of each strain gage to the center of the hole. Table 2 shows the values used in the experiment: Table 2. Values of R and Z i R(in) Z 1 (in) Z 2 (in) Z 3 (in).125.145.185.325 We have three unknowns, A, B, and C in Equation 1 this means three equations are required, which will be provided by installing three stain gages next to the hole. By solving the simultaneous equations we find the constants A, B, and C as bellow in terms of three measured strain, At the edge of the hole where maximum strain will be: the Where, S gi is the gage factor for the strain gages next to the hole and S g4 is the gage factor for the forth one. 5 3. Uncertainty in Simulation with COMSOL In order to find how errors propagate through the simulation we performed an uncertainty analysis using the Monte Carlo Method in our COMSOL model. The beam dimensions and load weight are model inputs with measurement uncertainty. Measuring the dimensions of the beam 1 times with the same device gives us the random uncertainty. The accuracy of the caliper used for measuring the dimensions of the beam and the hole, and the accuracy of scale used for measuring the weights are systematic uncertainties. After calculating the total uncertainty we run the Monte Carlo with mean and standard deviation (total uncertainty). For this purpose the parametric sweep function in the COMSOL comes very handy. For this specific model, COMSOL could reasonably handle running about 7 different cases for parametric study; so, to run 3 simulations in packs of 7 was quite time consuming. It is worth noting that an improvement of COMSOL s ability of running more cases in one simulation would be useful. 3.1 Parametric Sweep COMSOL has the power of taking variables in parametric format which is a great advantage for optimization and uncertainty studies in which we need to perturb many parameters. Parametric sweep is the key for this type of study. In this paper, we defined the geometry, material properties, and boundary conditions in parametric format to be able to change them as we wish for uncertainty analysis. We used Monte Carlo Method with 3 samples for our study. So, it is obvious that by having 3 different
cases it is almost impossible to create a model 3 times. In order to use parametric study we need to have the values of the first parameter in ascending or descending order. So, we defined a dummy parameter from 1 to 3 as the first parameter. The rest of the parameters can be in any arbitrary order. In order to find the random error for each variable, for example for linear dimensions we measured L, w, t, d, and lnom, 1 times with caliper; the average would be the mean value and the standard deviation is used to generate the random numbers. Similar method is used for finding the random values for the weights. But for the systematic error we look at the accuracy of the tool that was used. For example we used a caliper for measuring the linear dimensions which had accuracy of.1 inch for small dimensions and a different caliper for bigger dimensions with accuracy of.3 inch. It is worth noting that the systematic error is correlated for the dimensions that are measured with the same tool. For Modulus of elasticity, E, we tested the same beam in cantilever loading and we found E from the slope of the linear portion of stress versus strain (Hooke s law) and it was 1e6 psi, but the company that provides the Aluminum beams reports them as high strength aluminum; we expect high strength aluminum beam to have modulus of elasticity about 1.5e6 psi. So, we decided to use 1.4e6 psi as mean and 2.5% error for random uncertainty. The value of systematic error is considered zero for E. The random and systematic errors for each parameter is generated with random number generator in MS Excel and combined to give the total value of the parameters for 3 cases. The parameters used are showed in Table 3 and mean, random and systematic error for each parameter is tabulated in Table 4. Table 3. Parameters defined for the parametric sweep. dummy L w t d lnom P E Dummy variable in ascending order Length of the beam Width of the beam Thickness of the beam Diameter of the beam Distance, nominal stress to fixed end Load Modulus of Elasticity The number of decimal places for each variable is adjusted based on the instrument used for measuring that variable. For example we used a caliper for measuring the linear dimensions that reads up to three decimal places in inches. Table 4. The mean, and random and systematic error for each parameter used in COMSOL simulation. Parameter Average Random Systematic Error Error L 11.26.9.3 w 1.2.1.1 t.252.2.1 d.246.2.1 lnom 1.25.5.1 P1 4.1..1 P2.64.2.1 E 1.4E+7 2.5% 3.2 Post processing After running our model for the 3 Monte Carlo samples we need to process the data. The stress or strain concentration factor is found by dividing the max stress or strain by the nominal stress or strain. By defining a COMSOL max function and a cutline that goes through the center of the hole in the model we can easily get the maximum value of stress or strain. The nominal value of strain is found by using an average function and a cutline that passes through the nominal value of stress and strain. By using the node Derived Values and generating General Evaluation we get the average and nominal values in tabulated format. By dividing the maximum strain by nominal strain we find K t for all 3 cases. The last step is to calculate the average and standard deviation of K t and report it as stress concentration factor and total uncertainty in simulation. In Figure 6 and Figure 7 you can see the strain distribution through the hole and at the nominal strain location. In these figures, we have plotted the first 5 samples just to illustrate how COMSOL provides graphs for parametric sweep. Figure 8 illustrates the 3 dimensional graph of the stress distribution around the hole.
Figure 6. Strain distribution through the hole. relatively simple it may not be a problem if we run 3 Monte Carlo samples while fewer would be enough, but for a more complicated model it will certainly be essential to know when the number of samples converges. As seen in section 2, the number of samples for experiment converged about 15. Here, we look at the same graphs for the COMSOL Monte Carlo simulations. In Figure 9 you can see the average and standard deviation of K t plotted versus number of samples. The total uncertainty for K t in simulation was found.5 (one standard deviation). In Figure 1 the histogram of COMSOL simulated K t for 3 samples is compared to Gaussian distribution; note the K t histogram is not Gaussian. Figure 11 shows histogram of K t for experiment and simulation in one graph for better comparison. Figure 7. Strain distribution at nominal strain. Figure 8. Stress distribution in psi around the hole. 3.3 Convergence of number of samples Computing power is always a controlling factor in computational simulations. Thus, it is very important to know when the number of samples is enough, or in other words how it convergences. In this study since the model is 1.85 1.84 1.83 1.82 1.81 1.8 1.79 Convergence for Simulation 1 2 3 Number of Samples Average standard deviation.8.6.4.2. Figure 9. Convergence of average and standard deviation of Kt for COMSOL simulation. Number of Readings 6 4 2 Histogram of K t for Uncertainty in Simulation 1.598 1.636 1.674 1.712 1.75 1.788 1.826 1.863 1.91 1.939 1.977 2.15 2.53 Bin for K t Simulation Gaussian Distribution Figure 1. Histogram of k t for uncertainty in simulation plotted with comparison to Gaussian distribution.
Number of Readings 3 2 1 Figure 11. Overlapping histogram of K t from Experiment and Simulation for better comparison. 4. Results and Conclusions In this paper, we looked at experimentation, computational simulation and the uncertainty propagation. The stress concentration factor in experiment was found, 1.646 ±.141 and 1.646 ±.157 (from two different methods) and in simulation, 1.8 ±.5. Thus we have a comparison error 6 implying there is an unmodeled, un-simulated effect such as the strain gauge sensor glue. Or, perhaps there is an experimental or input uncertainty that is not captured. Parametric sweep in COMSOL is a simple tool to perform uncertainty analysis with Monte Carlo technique in our computational simulations. Additionally, this same method can be used to learn which parameter has more effect on the final result; in order to prevent consuming time and money on doing different experiments with different uncertainty in parameters, we can use uncertainty analysis in simulation to find out which parameter(s) are controlling factors. 5. References Histogram of K t for Uncertainty in Experiment & Simulation 1.584 1.619 1.654 1.689 1.725 1.76 1.795 1.831 1.866 1.91 1.937 1.972 2.7 2.42 Bin for K t Simulation Experiment 1. Hugh W. Coleman, W Glenn Steele, Experimentation, Validation, and Uncertainty Analysis for Engineers 3 rd Edition, page 64, John Wiley & Sons, Inc., Hoboken, New Jersey (29) 2. Instrumentation Division Measurements Group, INC, P-35 Digital Strain Indicator Instruction Manual, page 8 and 14. 3. Introduction to COMSOL Multiphysics Version 4.a (21) 4. S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, page 26, 27, 28, 65, 66, 67, 68, 9, 91, 92. McGraw-Hill Book Company, New York (197) 5. J. A. Gilbert, C. L. Carmen, MAE/CE 37 Mechanics of Materials Laboratory Manual Version 1., page numbers. Department of Mechanical and Aerospace Engineering, University of Alabama in Huntsville (2) 6. The American Society of Mechanical Engineers, ASME V&V 2-29 Standard for Verification and Validation in Computational Fluids and Heat Transfer, November 29 6. Appendix Figures Figure 12 andfigure 13 show the histograms used in for the Monte Carlo Method is the experimental uncertainty analysis and the simulation uncertainty analysis respectively. 35 3 25 2 15 1 5 35 3 25 2 15 1 5 ε1 496.113 499.443 52.774 56.14 59.434 512.764 516.95 519.425 522.755 35 3 25 2 15 1 5 47.379 41.798 414.218 417.637 421.56 424.475 427.894 431.314 434.733 Figure 12. Experimental uncertainty histograms. 4 3 2 1 4 3 2 1 35 3 25 2 15 1 5 ε3 364.319 367.83 371.342 374.853 378.365 381.877 385.388 388.9 392.411 L 11.224 11.233 11.242 11.251 11.26 11.269 11.278 11.287 11.296 t.242.245.247.249.252.254.257.259.261 E Bin 6.44E+1 6.63E+1 6.81E+1 6.99E+1 7.17E+1 7.35E+1 7.53E+1 7.71E+1 7.89E+1 35 3 25 2 15 1 5 1 8 6 4 2 5 4 3 2 1 Figure 13. Simulation uncertainty histograms. ε2 ε4 369.118 372.297 375.475 378.654 381.832 385.11 388.189 391.368 394.546 w.9965.9978.9992 1.5 1.18 1.31 1.44 1.58 1.71 lnom.999 1.4 1.1 1.15 1.21 1.26 1.31 1.37 1.42 6 5 4 3 2 1 35 3 25 2 15 1 5 d.236.238.241.243.246.248.251.253.255 P 17.769 17.857 17.946 18.34 18.123 18.212 18.3 18.389 18.477